Approximation in Sobolev spaces of nonlinear expressions involving the gradient

(1)

Ark. Mat., 40 (2002), 245-274

Approximation in Sobolev spaces of nonlinear expressions involving the gradient

P i o t r H a j t a s z a n d J a n M a l ~ ( 1 )

A b s t r a c t . We investigate a problem of approximation of a large class of nonlinear expressions

f(x, u, Vu),

including polyconvex functions. Here u: ~--+R m,

12CR n ,

is a mapping from the Sobolev space

W 1,p.

In particular, when

p=n,

we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in addition

n=rn,

satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation of

f(x, u, Vu)

are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions.

1. I n t r o d u c t i o n

W e a r e i n t e r e s t e d in a p p r o x i m a t i o n o f m a p p i n g s u: ~ - - - ~ R m f r o m t h e S o b o l e v s p a c e

WI,P(f~) n

b y a s e q u e n c e

{uk}k~__l

o f " m o r e r e g u l a r " m a p p i n g s t a k i n g c a r e o f t h e c o n v e r g e n c e

~ f(x, uk, Vuk)dx-+ ~ f ( x , u , V u ) d x

for a l a r g e c l a s s o f n o n l i n e a r i n t e g r a n d s .

H e r e a n d in w h a t f o l l o w s f~ C R n is a n o p e n s e t a n d m is t h e d i m e n s i o n o f t h e t a r g e t s p a c e for f u n c t i o n s c o n s i d e r e d . S o m e n o t a t i o n , t e r m i n o l o g y a n d c o n v e n t i o n s , m o s t l y s t a n d a r d , a r e e x p l a i n e d a t t h e b e g i n n i n g o f S e c t i o n 2 ( a l t h o u g h t h e y m a y b e u s e d b e f o r e t h e n ) so as n o t t o d i s t u r b t h e c o u r s e o f t h e i n t r o d u c t i o n t o o m u c h .

(1) The first author was supported by KBN grant no. 2-PO3A-055-14, and by a scholar- ship from the Swedish Institute. The second author was supported by Research Project CEZ J13/98113200007 and grants GA(~R 201/97/1161 and GAUK 170/99. This research originated during the stay of both authors at the Max-Planck Institute for Mathematics in the Sciences in Leipzig, 1998, and completed during their stay at the Mittag-Lefiter Institute, Djursholm, 1999.

They thank the institutes for the support and the hospitality.

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246 Piotr Hajtasz and Jan Mal~,

A standard way to approximate a function is by mollifying with a convolution kernel. T h e r e is also another familiar m e t h o d of approximation of uEWl'p(12), 1 < p < oc, by Lipschitz continuous (or even continuously differentiable) functions uk which yields both u k - ~ u in WI,p(12) and I{x:uk(x)~u(x)}l-+O. See e.g. [1], [5], [12], [23], [37], [43], [69] and also [24] and [39] for related approximations by Hblder continuous functions.

W h e n the two methods above fail, the approximation problem becomes difficult because of the lack of other powerful and elegant tools.

This is the case of approximation in nonlinear .A n spaces of functions with gradient minors in L pj spaces. This problem will be described in Section 3, but now for illustration we start with a very particular case.

Let u: f~-+R n be a mapping in the Sobolev class WI'P(ft) n, 1 ~ p < o c . Assume that in addition det VuELq(f~) for some p / n < q < o c . T h e LP-integrability of Vu implies only the LP/n-integrability of the Jacobian, so the L q integrability of the Jacobian is a strong additional condition. Now we ask:

Does there exist a sequence of "more regular" mappings uk: f~-+R n such that

(1) u k - + u in W l ' P ( ~ ) '~ and d e t V u k - - + d e t V u in Lq(~)?

T h e r e are many other related problems of approximation of determinants and minors, see e.g. [31, [11], [161, [18], [19], [22], [281, [38], [51], [52] and [53]. However very few results are in the positive direction. This was our main motivation for considering problems like (1).

It was not clarified here what we shall mean by "more regular" mappings. Ob- serve that if q = p / n , then we can get the desired approximation using the approximation by convolution and the resulting sequence {uk }k~_-i is C ~ smooth, so this is not a very interesting case and in what follows we shall assume that q>p/n. T h e n trying to approximate u by convolution or any other related m e t h o d we immediately lose information about the integrability of the Jacobian above the exponent p/n.

T h e problems with the approximation are caused by the high nonlinearity of the determinant.

We would be happy to have a sequence of smooth mappings, but, in general, when n - 1 ~ p < n it is not even possible to have a sequence of continuous mappings with the Luzin property (defined below); see Proposition 3.3. The counterexample is based on the fact t h a t the radial projection mapping x~-+x/]x[: B - + S c R n belongs to the Sobolev space W I ' p ( B ) n for all p < n . This forms a "topological" obstacle for the approximation. There are no such obvious obstacles when p = n , so one may hope to have a nice approximation. And indeed we prove t h a t when p = n and 1 _<q<oc there is a sequence {Uk}k~ as in (1) which consists of mappings which are

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Approximation in Sobolev spaces of nonlinear expressions involving the gradient 247 continuous, differentiable a.e. and have the Luzin property ((7), and L e m m a 1.2).

Although the case p = n was the main motivation for our research the m e t h o d applies also to the case p < n . Also then the approximating mappings are more regular t h a n generic mappings, but in a sense which is not so transparent and will be explained below.

In the case p < n there are alternative methods of approximation that will be explained in our forthcoming paper [25]. These methods are based on completely different ideas and the class of situations where they apply is neither wider nor narrower in comparison with the methods presented here.

Actually in the main theorem of the present paper, T h e o r e m 1.3, we succeed in approximating a given function uE WI'P(12) by a sequence {uk}~=l of locally weakly monotone functions (defined below) such that the approximating functions are "very close to u " - - s o close t h a t when the approximation is applied to each coordinate of the mapping uEWI,p(f~) m separately, surprisingly, nonlinear expressions involving the gradient (including the determinant) also converge. As we will see, locally weakly monotone functions are more regular than generic functions in WI'p(f~). If p ~ n , in fact, from the point of view of applications, their properties are almost as good as properties of Lipschitz mappings. To our knowledge, for such nonlinear problems that we consider, no natural approximation results were known at all.

Definition. Let uGWI,P(f~). We say that u obeys the weak maximum principle on 12 if the implication

( u - l ) + E W~'P(f~) ~ u < l a.e. in f~

holds for each 1 c R . Similarly we define the weak minimum principle. We say that u is weakly monotone if u satisfies b o t h the weak maximum and the weak minimum principles on all subdomains f/P~f~.

If there is R > 0 such that u is weakly monotone on all open subsets of ft of diameter less than or equal to R, then we say that u is locally weakly monotone.

We say t h a t a mapping u = (Ul ... Urn) E W l'p(f~)m is (locally) weakly monotone if each of the coordinate functions ui is (locally) weakly monotone.

Weakly monotone functions were introduced by Manfredi [44] (ef. [15], [33]

and [45]) as a generalization of monotone functions in the sense of Lebesgue [36], [50].

Locally weakly monotone functions obey some regularity properties t h a t we next describe.

L e m m a 1.1. Let uEWa,P(f~) m, 1 2 c R n, n - l < p < _ n be locally weakly mono- tone. Then u is locally bounded, differentiable a.e. and there exists a set Z with 7-ln-P(Z)=O, such that u is continuous at each point of f ~ \ Z .

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248 Piotr Hajtasz and J a n Mal~

Recall t h a t a Borel mapping u: 12-+R n, f l c R ", has the Luzin property if the image of any set of Lebesgue measure zero is of Lebesgue measure zero.

L e m m a 1.2. Let uEWl,n(12) m, ~ c F t n, be locally weakly monotone, then u is continuous and differentiable a.e. If re=n, u also has the Luzin property.

Remarks. 1. We refer to Section 4 for a more complete exposition of the theory of weakly monotone functions including proofs of the above two lemmas.

2. Here and in what follows, by differentiability we mean differentiability in the classical sense.

3. T h e Luzin property is very important as it allows one to apply the change of variables formula, see e.g. [6], [15], [21], [42], [47] and [56].

4. T h e m e t h o d of the proof works for uEWI'P(~) n, p>n, as well. Then, however, we do not gain anything interesting as every Sobolev mapping in WI'P(12) '~

is HSlder continuous, differentiable a.e. and has the Luzin property, see e.g. [6], [15]

and [46].

5. In the case p<n, (local) weak monotonicity does not imply continuity.

Indeed, the coordinate functions of the radial projection

x

u 0 : B )S, Uo(X)=lx I,

belong to the Sobolev space WI,p(B) for all p<n. T h e y are weakly monotone with the discontinuity at the origin. For other examples see [44].

6. L e m m a 1.1 does not extend to the case p < n - 1. Indeed, if ~E W 1'n-1 (Qn-1) is essentially discontinuous everywhere, then the function given by

~(xl, ..., x ~ - l , zn) = V ( X l , ..., xn-1)

is weakly monotone and essentially discontinuous everywhere in Qn.

Now we can formulate our main theorem.

T h e o r e m 1.3. Let uEWI,p(12), where ~cI:U ~ is open and l <p<oc. Then there is a sequence {Uk }k~_l of functions from WI'P(12) such that

(a) each uk is locally weakly monotone;

(b) Vuk=O a.e. in the set where uk#u;

(c) ~k-~u in w l , p ( ~ ) as k - ~ .

The essential novelty in the above approximation is the property (b). Observe that in the set where Uk=U we have Vuk=~Tu a.e., and thus we could write (b) in the equivalent form

Vuk = ~TuX{x:~k(z)=~(x)t.

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Approximation in Sobolev spaces of nonlinear expressions involving the gradient 249

Properties (b) and (c) do not necessarily imply t h a t [{X:Uk

(x) #u(x)

}l--+ 0. However, it follows from (b) and (c) t h a t , passing eventually to a subsequence,

(2)

X{x:uk(x)r

and Vu(x)~0} - ~ 0 a.e.

and

(3) X(x:Vuk(x)r a.e.

T h e conditions (b) and (c), and their consequences (2) and (3) are so strong t h a t t h e y imply unexpected convergence of nonlinear expressions involving the gradient in most of the interesting cases.

As we have already mentioned, weakly m o n o t o n e functions need not be continuous, and indeed, as we will see in Section 3, in general, when

n - l < p < n ,

it is not possible to find an a p p r o x i m a t i o n of u E W I'p(f~) by continuous functions t h a t would satisfy the conditions (b) and (c) at the same time (Corollary 3.4). This is in contrast with another method, mentioned above, which provides an approximation of

u E W 1,p

by Lipschitz, or even C 1, functions such t h a t

I{x:uk(x)~=u(x)}l--+O

and

uk--+u

in W 1,p, but, as we see now, the condition t h a t the gradient of

Uk

vanishes on the set where uk differs from u cannot be achieved.

Recently Vodop~yanov [65] found some applications of T h e o r e m 1.3 in the context of the change of variables formula on C a r n o t groups.

T h e m e t h o d of a p p r o x i m a t i o n in T h e o r e m 1.3 seems to be based on new ideas which are of independent interest. Some other ideas m a y however be traced back to the old work of Lebesgue [36] on m o n o t o n e functions. In order to construct locally weakly monotone functions we t r u n c a t e the function u infinitely m a n y times. After each truncation we obtain a b e t t e r function, looking more like a m o n o t o n e one.

Since we make infinitely m a n y truncations of a Sobolev function it requires quite delicate arguments to ensure t h a t the constructed sequence satisfies all the desired properties.

We promised to s t a t e our result as a result on a p p r o x i m a t i o n of a nonlinear expression involving the gradient. T h e way of convergence of

Uk

to U allows m a n y variants of convergence results for nonlinear integrands which m a y be easily derived by the reader from the properties (b) and (c). A p a r a d o x of the result consists in observing t h a t , although the t h e o r e m is scalar in nature, the whole strength a p p e a r s when applied coordinate-wise to vector-valued functions. Below we mention one result in the setting of functionals involving gradient minors. O t h e r results will be mentioned in Section 3.

l p

For a Wlo ~ -function u we define the

ith adjugate

adji V u as the collection of all i • i minors of Vu.

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250 P i o t r H a j t a s z a n d J a n Mal2~

Observe that for a mapping u: f~-+R m, f ~ c R n, the minors adj i Vu are defined only for l < i < m i n { n , m } . Obviously adjl V u = V u and if m = n , then adjn V u = det Vu and adjn_ 1 V u = a d j Vu.

The so called polyconvex integrals (see [2], [8], [9], [10], [17], [54] and [55]) arise in the context of nonlinear elasticity or in Skyrme's problem. Some typical polyconvex functionals lead naturally to a class of mappings u: f/--~R n with prescribed integrability of minors, i.e., adji VuE L p' (f/), where i= 1, 2, ..., n.

We may immediately state the result to cover the case of Orlicz type integrability.

T h e o r e m 1.4. Let uEWI,p(~'~) m, where f ~ C R n is open and l_<p<oc. Let g: ~2 x R m x R n m - ~ R be given by

g(x, ~, ~) = h(x, ~, ~)+al (X)Ol(ladjl r162 ~1),

where N = m i n { n , m}, (I)i: [0, oo)--+ [0, oc), i = 1, 2,..., N , are nondecreasing, al, ... ,

a N are nonnegative measurable functions on f~, and h: f~ • m x R n m - + R is a non- negative function such that the nonlinear operator

(4) u ~ - + h ( x , u , V u ) : W l , p ( ~ ) m ) Ll(l]) is well defined and continuous. Assume that

(5) f ]f~ g(x, u, Vu) dx < oc.

Then for any Carathdodory function f : f~ • R m • R a m - + R such that

(6)

there exists a sequence { Uk } C~= l C W l'p ( ~ ) rn of locally weakly monotone mappings such that

~ f (x, uk, V u k ) dx -+ ~ f (x, u, ~Yu) dx.

Remarks. 1. We will actually prove that the claim holds with a sequence

U oo

{ k}k= 1 as in Theorem 1.3. We want to emphasize that the approximating sequence

U oo

{ k}k=l is independent of the integrand f ( x , u, Vu).

2. The proof and applications of the theorem to more particular problems will be presented in Section 3.

3. A typical application is ~ i ( t ) = t p' , see Theorem 3.2.

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Approximation in Sobolev spaces of nonlinear expressions involving t h e gradient 251

4. We may consider some ai = 0 if we do not want to assume any integrability of the ith adjugate.

5. For a general theory of operators of the type (4), called Nemytskil operators, see e.g. [14] and [34]. However for most of the applications it suffices to assume t h a t h has a much simpler structure, for example h may be an integrable function independent of (~, ~), or h may equal [~1 p or [([q, where q is an embedding exponent, or

b(x)K[

q where

b 1/q

is a "multiplier" in the sense of Maz'ya and Shaposhnikova [48]

etc.

6. T h e most important case of the theorem is when

p=n,

since in the case

p<n

there are alternative methods of approximation of similar problems, see [25].

Now observe t h a t the theorem applies to problem (1). Indeed, assume t h a t u E W I ' p ( ~ - ~ ) n and that det

~'u(x)ELq(~t).

Then if we take

f ( X , ~, ~) : I~ --U(X)I p -~-I~ -- VU(X)I p + Idet ~ - d e t Vu(x)I q,

T h e o r e m 1.4 easily implies that there is a sequence of locally weakly monotone mappings

{uk}k~=l

CWI,P(12) n such t h a t

~ (IUk

- - u l P A r l V U k - V u l P " [ - Idet

VUk

- d e t

Vul q) dx

(7) O.

In particular when

p=n

we obtain an approximation by mappings which are continuous, differentiable a.e. and satisfy the Luzin condition (Lemma 1.2). This result is sharp as in the case n - 1

<_p<n

such an approximation is not possible in general

(Proposition 3.3).

The paper is organized as follows. The next section contains some auxiliary results needed in the sequel. In Section 3 we prove how T h e o r e m 1.4 follows from T h e o r e m 1.3, then we show some applications of T h e o r e m 1.4 to more particular situations and prove Proposition 3.3 and Corollary 3.4 demonstrating sharpness of the result on the Luzin property and of T h e o r e m 1.3 respectively. In Section 4 we recall the definition of the class of weakly monotone functions. We slightly improve known results and add some new observations. This will prove Lemmas 1.1 and 1.2.

Section 5 is devoted to the proof of the main result of our paper, Theorem 1.3.

2. N o t a t i o n and auxiliary results

T h e notation used in this paper is standard. The k-dimensional Hausdorff measure will be denoted by 74 k and the average value by

U E :

U d# -- I~(E) u d#.

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252 P i o t r H a j t a s z a n d J a n Mal~

T h e Lebesgue m e a s u r e of a set E will be d e n o t e d by [El. T h e characteristic function of t h e set E will be d e n o t e d by X E. T h e

L p

n o r m of a function u will be d e n o t e d by

[lull p.

T h e oscillation of a m e a s u r a b l e function over a m e a s u r a b l e set is defined a s

osc u = ess s u p u - ess inf u.

E E E

Positive a n d t h e negative p a r t s of a function u are

u + =max{u,

0}, u - = - min{u, 0}.

B y ~ we will always d e n o t e an o p e n subset of R n, even if n o t s t a t e d explicitly. T h e n u m b e r n will always d e n o t e t h e d i m e n s i o n of t h e E u c l i d e a n space in which we consider d o m a i n s of t h e studied functions.

B y

B = B ( x , r)

we will d e n o t e an n - d i m e n s i o n a l ball centered at x with ra- dius r. Spheres will be d e n o t e d by

S(x, r)=OB(x, r).

B y a C a r a t h ~ o d o r y function we m e a n a function f : ~ x R m x R n m - + R such t h a t f ( . , ~,~) is m e a s u r a b l e on for all ( ~ , ~ ) E R m x R ~m a n d

f(x, .,. )

is c o n t i n u o u s in R m x R nm for a l m o s t every x El2. B y C we will d e n o t e a general c o n s t a n t whose value is not i m p o r t a n t a n d so t h e s a m e s y m b o l C m a y d e n o t e different c o n s t a n t s even in t h e s a m e line. If we write

C=C(n,p)

we m e a n t h a t t h e c o n s t a n t C d e p e n d s on n a n d p only.

T h e gradient V u is u n d e r s t o o d in t h e distributional sense. Given l _ < p < c ~ , we d e n o t e by W I ' p ( D ) t h e usual Sobolev space on D consisting of t h e functions u such t h a t b o t h

uELP(12)

^{a n d}

IVulELP(D).

T h e space is equipped w i t h t h e n o r m [[U[[1,p=[lU[[p~-II~Tu[[ p. B y

W~'P(12)

we will d e n o t e t h e closure of C~r in t h e n o r m [[ 9 [[1,p. B y

WI'p(12) m

we d e n o t e t h e class of m a p p i n g s u: 12---~R "~ such t h a t t h e c o o r d i n a t e functions belong to t h e function space W I ' p ( ~ ) . A similar c o n v e n t i o n is used also for o t h e r function spaces.

N o w we collect some s t a n d a r d results t h a t will be used in t h e sequel. We suggest t h a t t h e reader skip reading this section a n d j u m p to Section 3. T h e n t h e reader c a n consult Section 2 w h e n e v e r necessary.

T h e following l e m m a on differentiation of absolutely c o n t i n u o u s measures c a n be found e.g. in [43, C o r o l l a r y 1.19] or in [15, P r o p o s i t i o n 4.37]. It is a consequence of a s t a n d a r d covering a r g u m e n t .

L e m m a 2 . 1 .

If vEL~oc(~t), then

7-l~-P({xEgt:limsuprP~ ( I v l d x > O } ) = O .

r-+O J B ( x , r )

T h e following result is due to S t e p a n o v , see [41], [60] a n d [61].

L e m m a 2 . 2 .

A function u: D-+ R is diI~erentiable a.e. if and only if lu(y)-u(x)l

lim sup < oc a.e.

y-~x ]y-x]

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Approximation in Sobolev spaces of nonlinear expressions involving the gradient 253 T h e next results concern properties of functions from the Sobolev space

Wl'P(~t).

First we recall that b o t h WI,P(gt) and WJ'P(~t) have the

lattice prop- erty,

i.e. they are closed under taking pointwise m a x and min. For the following result, see e.g. [27, L e m m a 1.25].

L e m m a 2.3.

IfuEWI,P(~), vEWJ'P(~) and

]ui_<iv I

a.e., then

uEW01'P(~).

Proof.

Assume first that 0 < u < v . Let ~ k E C ~ ( F t ) be such that

~ak--~v

in

WI'P(~).

T h e n the functions min{~k,u} have compact support and it is easy to prove t h a t min{~k, u}-+min{v,

u}=u,

so

uEWJ'P(~).

Now we pass to the general case

lul<iv].

It is easy to see that

v+,v-EWI'p(~).

Hence

Ivl=v++v-E WJ'P(~).

Now the inequalities 0~u+_< I v] imply t h a t u +, u-EWJ'P(~t). Hence also u = u + - u - eW~'p(a). []

C o r o l l a r y 2.4.

Let u, vEWI,P(~) and EC~. If

0 < u < v

a.e. on E and vXEE W~'P(~), then uX~EW~'P(~).

Proof.

The claim follows from the previous lemma and the observation t h a t

UXE=min{u+,VXE}EWI'P(~). []

L e m m a 2.5.

Assume that

u E W I , P ( ~ )

and

t > 0 .

Then uEW~'P(~) if and only if

min{u,

t}EW~'P(12).

Proof.

T h e implication from the left to the right follows from L e m m a 2.3.

Suppose now that min{u, t } E W I ' p ( ~ ) . Since u----rain{u, t } - EW~'P(~) it remains to prove t h a t

u+EW~'P(12)

and thus we can assume that u__0. For each positive integer k, the function

ktu u k - kt+u

is in

WI,P(~)

by the chain rule. Since

uk <_k

rain{u, t}, appealing to L e m m a 2.3, we have t h a t

ukEWI"P(~).

A routine argument shows that u belongs to

W~'P(12) as

the limit of the sequence {uk}~_l- []

We need also consider infima and suprema of infinite families of Sobolev functions. If/4 is a family of measurable functions on ~t, we define the lattice supremum V/4 as the supremum with respect to the ordering, neglecting sets of measure zero.

Thus, V/4 is an a.e. majorant of each element o f / 4 and an a.e. minorant of each a.e. majorant of/4. Similarly we introduce the lattice infimum A/4.

It will be convenient for us to allow the lattice supremum to be + o c and the lattice infimum to he - c o on sets of positive measure.

I f / 4 is a countable family, then V/4 can be obtained as the pointwise supremum of/4. However, i f / 4 is uncountable, we must distinguish between the lattice

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254 Piotr Hajtasz a n d J a n Mal~

supremum V/4 and the pointwise supremum

sup/4: x, >sup{u(x):uE/4}.

T h e latter one heavily depends on the choice of representatives.

T h e following accessibility property can be found e.g. in [49, L e m m a 2.6.1]. For the sake of completeness we provide a short proof.

L e m m a 2.6. Let/4 be a class of measurable functions defined in a measurable set E c R u. Then V/4 exists and there is a countable subfamily VC/4 such that

V/4 -- V ~; -- sup ]).

Proof. First observe t h a t we may assume that the family /4 is bounded in L ~ and consists of nonnegative functions, otherwise we replace/4 by a family of functions 89 where uE/4. We can aJso assume t h a t the functions are defined in a set of finite measure, otherwise we make a diffeomorphic change of variables which maps E onto a bounded set. Let

S : s u p { / E m a X { U l , . . . , u k } d X : u l , . . . , U k E / 4 for some k } <(x).

Now there exists a sequence {uk}~=lC/4 such that v k = m a x { u l , . . . , u k } satisfies

V c~

limk-,cr fE Vk dx=s. Since { k}k=l is nondecreasing we have the a.e. convergence l i m k - , ~ vk =v. Obviously fE V dx=s. This easily implies that v--V/4 and so we can take ] ; - - { u l , u 2 , . . . } . []

Now, we mention a folklore theorem on suprema of infinite families of Sobolev functions.

L e m m a 2.7. Let /4CWI'P(~~) be bounded (in the Sobolev norm) and closed under finite maxima. Then V / 4 E W I ' p ( ~ ) . Moreover V/4 is a pointwise limit of an increasing sequence of functions from/4.

Proof. Let u=V/4. Then, using t h a t / 4 is closed under finite maxima, we infer from L e m m a 2.6 t h a t there is an (a.e.) increasing sequence {Uk}~=l of functions from /4 such that Uk-+u a.e. T h e sequence {Uk}k~=l is bounded in W~'P(~), so that by [27, T h e o r e m 1.32], uEW01'P(f~) and in fact u is a weak limit of {Uk}k~=l in

[]

We shall need the following elementary Poincar~ lemma, see e.g. [12] or [69].

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L e m m a 2.8.

If uG WI'p(B(x, r)), then fs(x,~) luI p dx~-CrP fB(,,~) IVuIP dx.

The next lemma is a direct consequence of the Sobolev embedding theorem into the space of HSlder continuous functions. It is often called Gehring's oscillation lemma. For a proof see e.g. [43, Lemma 2.10].

L e m m a 2.9.

If uEWl,P(S(xo,t)), p > n - 1 , then u is H61der continuous and ose u ~ C ( n , p ) t ( / s IVuiPdT-ln-1) 1/p.

S(~o,t) (xo,t)

The following two lemmas are special cases of more general results. The proof of the first lemma can be found in [35, Section 8.3.3].

L e m m a 2.10.

If

~ c R n

is a bounded domain with a smooth boundary, then there exists a bounded linear extension operator E: W l'n- l ( Ol~ ) --+ W l'n ( ~ ) with the additional properties that E u E C ~ ( ~ ) for any u E W I ' n - I ( o ~ ) and E u E C ( ~ ) for any uEC(OI2)NW 1,~-1 (0~).

The proof of the next lemma can be found in [13], [15, Theorem 5.6], [19]

and [21].

L e r n m a 2.11.

If uE W l'l ( B ) n is continuous and satisfies the Luzin property, then

[u(S)l -~ ]B [det ~Tu I

dx.

The last lemma is proved in [54, Theorem 3.2].

L e m m a 2.12.

If

vEWI'P(~'~) n

and ]adjVviELq(12), p > n - 1 , q > n / ( n - 1 ) , then d(vl dv 2 A...Adv n)-=dv I Adv2 A...Adv n in the sense of distributions, i.e. for eve- ry ~ e C ~ ( ~ ) the following identity is true

- ~ vl d~zAdv2 A...Adv'~= ~ Cdvl Adv2 A...Adv n.

3. Convergence o f nonlinear expressions

In this section we prove the assertions mentioned in the introduction concerning convergence of expressions of the type

~ f ( x ,

^u,Vu) dx.

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256 P i o t r H a j l a s z a n d J a n Mal~

First we will show how to obtain T h e o r e m 1.4 as a consequence of T h e o r e m 1.3.

T h e n we will provide some applications to more particular problems. T h e proofs in this section are not hard, because the deep step is done in T h e o r e m 1.3. T h e proof of T h e o r e m 1.3 is p o s t p o n e d to Section 5.

?~ oo

Proof of Theorem

1.4. Assume t h a t

uEWI,p(f}) m

satisfies (5). Let {

k}k=l

be an a p p r o x i m a t i n g sequence as in T h e o r e m 1.3. More precisely {uk}k~__l is obtained by applying T h e o r e m 1.3 to each coordinate of u separately. Let us write

~k(x) = f(~, u~(z), vuk(~)), ~(~) = f(~, u(x), w(~)), Ck(x) =g(~, uk(z), Vuk(~)), ~(~) =g(~, u(x), w(~)),

~k(x) = h(z, uk(~),

Wk(X)),

~(Z) = h(x,

~(~), Vu(x)).

First we notice t h a t if we denote the i t h coordinate by a superscript, each Vu~ is either 0 or U u i. This has the consequence t h a t each gradient minor of uk either vanish or is equal to the corresponding minor for u in a given point. Hence

[adjj

Vuk[ <_

[adjj Vu[, and since ~ j is nondecreasing, also

~y([adjj Vuk[) _< (Pj ([adjj Vu[).

It follows t h a t

(s)

and thus Hence

Ck -< r

I~k-~l -< r162 < ~k+2r < 1~k-71+3r

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~ l~k--~l dx << ~ lTlk--711dx+ ~ min{l~k--qoh 3r dx.

T h e first integral on the right tends to zero by the assumptions on h. T h e convergence of the second integral to zero follows from the Lebesgue d o m i n a t e d con-

O O

vergence t h e o r e m as soon as we show t h a t from any subsequence of

{~k}k=l

we can e x t r a c t a subsequence t h a t converges to ~ a.e. This follows from the fact t h a t

uk-+u

in W I'p and t h a t f is a C a r a t h ~ o d o r y function. []

Although T h e o r e m 1.4 is very general, it does not cover all possible applications of a p p r o x i m a t i o n from T h e o r e m 1.3. We state here one typical consequence of T h e o r e m 1.3 which is of a slightly different nature. It is based on a direct application of the fact t h a t for u E W I , p ( ~ ) n either

Vuk(x)=Vu(x) and Uk(X)=U(X),

or

detVuk(x)=O.

This allows one to have no growth condition for the integrand depending on (x, ~, ~) outside the set where det ~ = 0 .

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C o r o l l a r y 3.1. Let uEWI'P(f~) n and let f : f~ x R n • be any function

U oo

such that f ( . , . , O ) = O and f ( x , u ( x ) , d e t Vu(x))ELq(f~), l < q < o o . Let { k}k=l be an approximating sequence satisfying the properties (b) and (c) of Theorem 1.3 coordinate-wise. Then we have

f l f ( x , u ( x ) , d e t V u ( x ) ) - f ( x , u k ( x ) , d e t V u k ( x ) ) l q d x ~ O , as k--+c~.

Proof. The integrand equals

If(x, u(x),

det Vu(x))l q multiplied by a characteristic function of the set {X:Uk(X)~U(X) and ~7u(x)5~0}. Hence the convergence follows from (2) and the Lebesgue dominated convergence theorem. []

Now we state the most typical application in terms of nonlinear function spaces introduced by Ball [2], see also [17].

Definition. Let P--(Pl,-.. ,PN) be a multi-index, where l < p l < c ~ , O<_pj<oo, j = 2 , 3 , . . . , N , and N = m i n { m , n } . We denote by Ap(f~) the class of mappings u E W l ' m ( f t ) m such that ladjj ITulELP3 for all j E { 1 , . . . , N } with p j ~ 0 . This, in particular, means t h a t we pose no assumptions about the integrability of the j t h adjugate when pj =0. We will say that uk approximates u in A n or that uk--+u in A n, if uk--+u in W I , p l ( ~ ' ~ ) m and ]axijj •Uk--m:tjj Vu]--+O in LPJ (gt) for all j E { 1 .... , g } with pj 50.

As special cases we consider the John Ball class Ap,q with p=(p, O, ..., O, q, 0), see [2], [64] and [54], and spaces Bp,q with p=(p, O, ..., O, q), both for g = m = n and

l <p,q<oo.

Thus the John Ball class Ap,q(f~) consists of all mappings in w l , p ( f t ) n such t h a t adjVuELq(f~), l<p,q<c~, and the space Bp,q(f~) of the mappings u e W I ' P ( ~ - ~ ) n

with det V u E Lq(f~), l <p, q< cxD.

T h e o r e m 3.2. Let uE.Ap(f~) and let p be a multi-index with l < p l < c ~ and

U oo

0 < p j < e o , j = 2 , . . . , N . Let { k}k=l be an approximating sequence satisfying the properties (b) and (c) of Theorem 1.3 coordinate-wise. Then Uk--+u in .Ap(f~).

Proof. We set

N

f ( x , ~, ~) =

IC-u(z)p ~ +~-~ ay ladjj ~ - adj~ Vu(x)l p~,

j = l where

aj = ^1, pj > 0, O, pj-~O.

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258 Piotr Hajtasz and Jan Mal)~

If we let qj =max{pj, 1}, we have

ladjj ~ - a d j j Vu(x)] pj _< 2q~-1 (ladjj ~1 p~ + ]adjj Vu(x)l p~ ).

We see t h a t the assumptions (5) and (6) of Theorem 1.4 are satisfied with

N

h(x, ~,~) = I~-u(x)l pl

~-~

aj2qJ-lladjj Vu(x)l pj and Cj(t) = 2qJ-ltP'.

j = l

Hence the convergence Uk--+u in Ap(f~) follows directly from T h e o r e m 1.4. []

Let l < q < c c . Recall t h a t by T h e o r e m 1.3, L e m m a 1.2 and T h e o r e m 3.2, each mapping uEBn,q(~l) can be approximated in B,~,q by a sequence {Uk}k~=l of continuous mappings with the Luzin property. We are going to show that n is a borderline exponent for such approximation. This shows the sharpness of our results.

P r o p o s i t i o n 3.3. The radial projection mapping u: B(O, 1)-+S(0, 1) given by u(x)=x/]x I belongs to Bp,q(B) for all l < p < n and l < q < c ~ . If n - l < p < n and l < q < c ~ , then u cannot be approximated by continuous mappings uk E Bp,q( B ) with the Luzin property in the metric of Bp,q. In particular Uk cannot be Lipschitz continuous.

Proof. We argue by contradiction. Let n - l < p < n , l_<q<c~ and suppose that a sequence {uk}~=l C Bp,q(B) of continuous mappings with the Luzin property converges to u in the metric of Bp,q. T h e n applying a version of the Fubini theorem valid for Sobolev spaces we conclude that for almost all r E ( 0 , 1) after taking a subsequence we have the convergence

Ukls(O,r)-+Uls(O,r) in WI'P(S(O,r)),

where the restrictions to spheres are understood in the sense of traces. Fix one such r E (0, 1). By L e m m a 2.10 there exists a bounded linear extension operator

E: WI'n-I(OB(O, r)UOB(O, 1)) ~ ~ Wl'n(B(O, 1 ) \ B ( 0 , r)) n such that Ev is smooth in B(0, 1 ) \ B ( 0 , r). Let

f 0 on OB(O, 1),

V k

!

_{u k - u} _on_OB(O,r),

and define

u + E v k o n B ( O , 1 ) \ B ( O , r ) ,

W k uk o n / ~ ( 0 , r).

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It is easy to see t h a t w~: B--+R '~ is continuous and identity on the boundary. Hence by the Brouwer fixed point theorem

BCwk(B),

and thus

]wk(B)l>]B [.

Moreover, w k ~ W I ' p ( B ) n satisfies the Luzin property (the Luzin property is satisfied in the annulus B(0, 1)\/3(0, r) because wk is smooth there). Hence invoking Lemma 2.11 we obtain

IBt<_Iwk(B)I<_/B IdetVwkldx

(0,1)

=/B IdetVukldx+ f "ldetVwkldx"

(O,r) JB(O,1)\B(O,r)

The second integral in the right-hand side converges to zero. Indeed,

vk--->O

in

WI'n-I(OB(O, 1)UOB(O,r)),

so

wk=u+Evk-+u

in

Wt,n(B(O,

1)\B(0, r)) which implies t h a t f f [det

Vwk[ dx --> ff

[dee Vu[

dx

JB

( 0 , 1 ) \ S ( 0 : r ) J B(O,1)kB(O,r)

Hence

(10) lim inf f ]det Vuk[

dx > [B I

k--~oo JB(O,r)

which contradicts the convergence det Vuk--+det Vu--0 in L q. []

Remarks.

1. When

n - l < p < n

the proof can be slightly simplified. Indeed, by the Sobolev embedding theorem

uk--+u

uniformly on the sphere S(0, r). Then the construction invoking the extension operator is not needed as inequality (10) follows rather easily from the Brouwer theorem.

2. The obstacle for the existence of the approximation has a topological nature, it is essential t h a t we create a hole in the image of the mapping. If

p=n,

then we cannot construct counterexamples as above simply because u:

x~-+x/Ix[

does not belong to WI:n(B)'L

C o r o l l a r y 3.4.

The function ui(x)=xi/IxIEWl'p(B),

iE{1, ... ,n},

n - l < p <

n, cannot be approximated by a sequence of continuous functions which satisfy the conditions

(b)

and

(c)

of Theorem

1.3.

Proof.

We argue by contradiction. If there is such a sequence for one i, then it can be found for each i and thus there is an approximating sequence

{vk}k=t

oo for

u(x)=x/lx I

satisfying the properties (b) and (c) coordinate-wise. Since Vv~(x)E {Vui(x),0} a.e., it follows that the functions vk are locally Lipschitz continuous outside the origin and thus satisfy the Luzin property. Hence by Corollary 3.1, vk approximate u in

Bp,q

which contradicts Proposition 3.3. []

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260 Piotr Hajlasz and J a n Mal)~

4. W e a k l y m o n o t o n e f u n c t i o n s

We say that

uEWI'P(~'~),

l<p<cx~, is weakly K-pseudomonotone, K > I , if for every x E ~ and a.e. 0 < r < d i s t ( x , ~c),

osc u < K osc u, B(=,r) S(z,r)

where the oscillation on the left is essential with respect to the Lebesgue measure and the oscillation on the right is essential with respect to the (n-1)-dimensional Hausdorff measure.

This class contains a class of weakly monotone functions, see Section 1. The proof of the fact t h a t weakly monotone functions are weakly 1-pseudomonotone is standard and left to the reader. One may, e.g., use the characterization of the Sobolev space by the absolute continuity on lines, see [12, Section 4.9.2] and [43,Theorem 1.41].

The following result is a slightly stronger version of a result due to Manfredi [44]

(cf. [64]).

T h e o r e m 4.1. Let uE W~lo'v( ~ ) be weakly K-pseudomonotone for some K>_ 1.

(1) If n - l <p<n, then u is locally bounded and

(11) ( osc u ~ < C ( n , p ) K V r P ~ IVulPdx,

~ , B ( ~ o , r ) - ( = o , 2 , ' )

whenever B(xo,2r)C~. Moreover there exists a set ZCI2 with 7-/n-P(Z)=0 and such that u is continuous on the set 12\Z.

(2) If p=n, then u is continuous. Moreover

(12) ( osc u ) n < C ( n ) K n / B IVu[ ndx,

~B(=o,r) - log(R/r) (~o,a) whenever B(xo, R ) C ~ and r <R.

Remark. The estimate (12) is a generalization of the Courant-Lebesgue lemma, see [7] and [31, Lemma 8.3.5].

Proof. Let uEWl'V(12), p > n - 1 , be a weakly K-pseudomonotone function, and let B(xo,R)C~2, r < R . By Fubini's theorem, uEWl'p(S(xo,t)) for almost all r < t < R . Hence by Lemma 2.9 we get a family of inequalities

(/S /\lip

(13) osc u < C(n,p)t IVul p dT-I n - i } 9

S(=o,t) (=o,t)

(17)

Integrating those inequalities with respect to t yields

(14) ( osc u ~ dt < C(n,p) ]VuiP dx.

\S(xo,t) / t P - - n T 1 - (xo,R)\B(xo,r)

Since u is weakly K-pseudomonotone we get

(15) osc u < osc u < K osc u

B(xo,r) B(xo,t) S(xo,t)

for almost all t E Jr, R].

From (14) and (15) we infer (11) and (12). Inequality (12) implies the continuity when p=n. T h e continuity outside a set Z with 7-ln-P(Z) =0 is a direct consequence of inequality (11) and L e m m a 2.1. []

Remark. In the case n - l < p < n , Manfredi [44] and also Sver~k [64] obtained a weaker result. T h e y proved continuity of u outside a set with vanishing p-capacity.

The estimate was improved in [55, Theorem 7.4].

Yet another property of weakly K-pseudomonotone mappings has been obtained by Mal~ and Martio [42], see also [40, Theorems 3.4 and 4.3].

T h e o r e m 4.2. If uEWl'n(12) n is weakly K-pseudomonotone for some K > 1, then u has the Luzin property.

T h e following result is known among specialists as folklore. For the case p--n see [58, Corollary, p. 341] and also [40], the general case can be found in [66].

T h e o r e m 4.3. I f u E Wlo' c (~), l p p > n - 1, is weakly K-pseudomonotone for some K>_I, then u is differentiable a.e.

Proof. I f p > n , then any Sobolev function is differentiable a.e., so we can assume t h a t p<n. We also can assume that p<n, since Wllo'~ C Wllo 'p for p<n. Then by (11) the condition from the Stepanov theorem (Lemma 2.2) is satisfied, whenever x is a Lebesgue point of [Vu[ p. []

Observe that Lemmas 1.1 and 1.2 are consequences of the above results.

We close this section by recalling two classes of examples of weakly monotone functions.

Definition. We follow Koskela, Manfredi and Villamor [33]. Let 0<a(x)_</~<

c~ a.e. in ~, where ~ is a measurable function a n d / ~ is a constant. Let l < p < c ~ and .A: 12 • R n - + R n be a Carath~odory function such that

]A(x,~)]</~]~l p-1 and A(x,~).~>_a(x)]~] p.

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262 Piotr Hajlasz and Jan Mal)}

Weak solutions u E W, lo~(D) to the equation div A(x, V u ) = 0 , are called A-harmonic functions. The usual assumption is t h a t a ( x ) > a > 0 a.e., but our assumption is much weaker, so the equation may be very degenerate. It is easy to prove, [33], t h a t A-harmonic functions are weakly monotone. Thus the properties of weakly monotone functions yield the following result.

T h e o r e m 4.4. Let,4 be as above, n - 1 <p<_n. Then any A-harmonic function u is locally bounded and continuous outside a set Z with 7-ln-P(Z)=0. Moreover u is differentiable a.e.

Remarks. 1. The continuity property has been observed in [33], but with a worse estimate for the size of the discontinuity set. The differentiability result generalizes that of Sojarski [4], and Reshetnyak [57]; see also [26], [30] and [32] for related elliptic results, and [62] and [63] for related parabolic results.

2. In the nondegenerate case ( ~ ( x ) > a > 0 a.e., it is known that any A-harmonic function is Hhtder continuous, [43], and differentiable a.e., see [4] and [57].

Definition. We say that the mapping uE WI'P(~) ~, 12CR n, has finite dilatation if there is a function K , l < _ K ( x ) < c c a.e., such t h a t

IVu(x)V ~ < K(x) det Vu(x) a.e.

In other words finite dilatation means t h a t for almost every point x either det Vu > 0 or V u ( x ) = 0 .

Obviously mappings with det V u > 0 a.e. have finite dilatation.

Gol'dshteYn and Vodop~yanov [20], proved that if p=n, then the coordinate functions of a mapping with finite dilatation are weakly monotone. The following result is a generalization of the result of Gol'dshte~n and Vodop'yanov [20], see also [15], [29], [44], [54] and [64].

T h e o r e m 4.5. If

uEAp,q(~),

p > n - 1 , q > n / ( n - 1 ) , is a mapping of finite dilatation, then the coordinate functions of u are weakly monotone.

Proof. We argue by contradiction. Suppose that one of the coordinate functions, say u 1, is not weakly monotone. Then there is l~'~!2 and m, M E R such that either ( u l - M ) + e W ~ ' P ( ~ ' ) and I{xel2':ul(x)>M}l>O, or ( m - u l ) + e W ~ ' P ( ~ ' ) and I{xe~':ul(x)<m}l>O. Assume the first case. Let f i l = m i n { u l , M } X ~ , + ulX~\n, and ~ - - ( f i l , u 2 , . . . , u n ) . Obviously fiEAp,q(12). We will prove in a while that

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Before we do this, however, we show how to complete the proof of the theorem. Let

E = { x E f Y : u l ( x ) > M } .

Observe t h a t det V f i = 0 in E. This, identity (16) and the fact t h a t u = f i in f ~ \ E imply t h a t

E d e t V u

dx = / E

det Vfi

dx = O.

Hence the finiteness of the dilatation implies t h a t V u - - 0 in E which is not possible.

T h u s we are left with the proof of (16).

Let C E C ~ ( ~ ) such t h a t ~bl~,-1. Employing L e m m a 2.12 we obtain

/

(det V u - d e t Vfi)

dx

= f ~ r V u - d e t Vfi)

dx

= f~ r Adu2A...Adu n -dfi 1 Adu2A... Adu n)

= - ~(ul-ftl)dCAdu2A...Adun=O. [3

5. T h e p r o o f o f t h e m a i n r e s u l t

This section is devoted to the proof of T h e o r e m 1.3. T h e proof of the theorem is quite difficult so we s t a r t with describing the main idea. We hope it will help to understand the steps of the proof.

T h e rough idea is the following. Applying infinitely m a n y corrections "from above" to u we make the function "locally weakly u p p e r monotone". T h e n applying infinitely m a n y corrections "from below" we make it "locally weakly lower monotone". T h e resulting function is locally weakly monotone. Now we describe the corrections "from above". T h e function fails to satisfy the weak m a x i m u m principle on an open set f~'~12 if there is a t E R such t h a t

(u-t)+EWI'p(~ ')

a n d

u>t

on some subset of ~ with positive Lebesgue measure.

This suggests the m e t h o d of corrections "from above". We fix R > 0 . Whenever

ECl2

is a set with diameter less t h a n or equal to R and such t h a t

u>t

a.e. in E and

(u-t)XEEWI'P(~)

we replace u with the

upper truncation v-~tXE-~-U)(~]\E.

T h e

upper R-correction

is defined as the infimum of all u p p e r truncations over all t real and E as above. We prove then t h a t the resulting function satisfies the weak m a x i m u m principle on all open sets 1 2 ~ l] with d i a m e t e r less t h a n or equal to R.

This is a delicate construction as a priori we take the infimum over an uncountable set of functions. Since each function is measurable, and hence defined up to a

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264 Piotr Hajtasz and Jan Mal~

set of m e a s u r e zero, such an infimum does n o t really make sense. T h u s we have t o be very careful in our constructions.

T h e n we a p p l y similar corrections "from below" t o t h e modified function to m a k e t h e function satisfying t h e weak m i n i m u m principles on o p e n sets 12'~ ~ w i t h d i a m e t e r less t h a n or equal t o R. T h e resulting function is locally weakly m o n o t o n e . N e x t we prove t h a t passing to t h e limit as R--+0 gives t h e desired a p p r o x i m a - tion.

Observe t h a t in each step of t h e correction we get a function v w i t h Vv=O on t h e set {x:u(x)~v(x)}. This p r o p e r t y is stable w h e n we take t h e infimum a n d hence p r o p e r t y (b) follows.

To prove (c) we use t h e fact t h a t t h e LP-distance of t h e R - c o r r e c t i o n from u is limited by t h e scaling c o n s t a n t in t h e Poincar~ inequality which behaves as R.

T h e p r o o f is quite long a n d difficult, so we divide it into several lemmas.

It is c u s t o m a r y to call t h e s y m b o l s U a n d M, c u p a n d cap, respectively. T h e s h a p e of t h e s y m b o l s explains t h e t e r m i n o l o g y t h a t follows.

First we define t h e class o f sets on which we will t r u n c a t e t h e f u n c t i o n u.

Let vEWI'P(~) a n d a E R . We say t h a t a Borel set E c l 2 is an a-cap set (a-cup set) for v if v>a ( v < a ) a.e. on E a n d (v--a)XEEW~'P(fl). We say t h a t E is a cap set (cup set) for v if E is an a - c a p set (a-cup set) for v a n d some a E R .

Let E be an a - c a p set or an a - c u p set of positive m e a s u r e for v. Since no characteristic function o f a set o f positive m e a s u r e c a n be in Wol'P(fl), t h e value a is uniquely d e t e r m i n e d by the function v a n d t h e set E . (In fact, it is t h e essential infimum (for a c a p set) or essential s u p r e m u m (for a c u p set) of v over E . ) We set

v = ( v - a ) % .

We say t h a t v is R-capless (R-cupless) if all c a p (cup) sets for v of d i a m e t e r less t h a n or equal t o R have m e a s u r e 0.

L e m m a 5 . 1 . If uEWI,P(12) is R-capless and 1 2 ' ~ with diaml2~_<R, then u satisfies the weak maximum principle on ~ . If u is R-cupless and ~ 1 2 with d i a m l~' < R, then u satisfies the weak minimum principle on ~'. Hence any function u that is both R-capless and R-cupless is locally weakly monotone.

Proof. A s s u m e t h a t uEWI'P(~) is R-capless. Suppose t h a t 1 2 ~ is an o p e n set w i t h d i a m ~ ' < R a n d ( u - t ) +

EWI'P(~~').

I f E = { x E ~ ' : u ( x ) > t } , t h e n ( u - - t ) X E = (u--t)+X~,EW~'P(~) and t h u s E is a t-cap set for u. Since u is R-capless, I E I - - 0 a n d t h u s u<t a.e. on 12 ~. This verifies t h e weak m a x i m u m principle on ~ . Similarly we can prove t h e s t a t e m e n t a b o u t t h e weak m i n i m u m principle a n d t h e final p a r t is obvious. []

(21)

T h e last l e m m a suggests t h a t to m a k e the a p p r o p r i a t e corrections, the right idea will be to "remove caps and cups".

First we need to prove some nice properties of cap sets and truncations.

L e m m a 5.2. Let vEWI,p(12). I r E is an a-cap set for v, F is a b-cap set for v and a<_b, then F \ E is a b-cap set for v. Hence max{v E, ~)F} : m ~ x { v E, v F \ E } _ ~ V E -~-vF\ E "

Proof. T h e first assertion follows from the identity ( v - b ) X F \ E = v F - m i n { v F, vE}.

T h e second one is an obvious consequence. []

L e m m a 5.3. Let vEWI,p(f~), R > 0 , g be a subclass of the family of all cap sets E for v with d i a m E ~ R and

Then ~ e W o ~ , ' ( a ) ,

(17) and

w = V { v E : E E C}.

V w = VvX{x:w(x)>0}

(is) Ilwll, <_ CRIIVvlI~.

Proof. First let us assume t h a t g is a finite family. T h e n we use L e m m a 5.2 to show t h a t we m a y pass to a finite disjointed family ~" of cap sets F for v with diam F_< R, such t h a t

w = V l v F : F E ~ ' } = ~ v F.

FE~=

This proves t h a t wEW~'P(f~) and (17) holds. Fix F e ~ ' . T h e n there is a ball B ( z , R) in R n which contains F. T h e function v F can be extended to a function in W~'P(B(z, R)) by setting v F = 0 outside f~. By the Poincar6 inequality ( L e m m a 2.8) we have

S u m m i n g over F E 9 r we obtain

~l

^{wl ~}dx <_ C R p

~

I V v l ~ dx.

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266 P i o t r H a j t a s z a n d J a n Mal~

Now we consider the general case of s Let

/4 = {max{v E1 .... , v Ek } : E l , ..., Ek E C for some k}.

It follows from formulas (17) and (18), proved above for finite m a x i m a , t h a t U is bounded in W~)'P(~). Hence L e m m a 2.7 implies t h a t w-~VI4EW(~'P(I~) and there is a countable subfamily {El, E2, ...} C ~ such t h a t {wk}k~__l,

W k : I n a x { v E 1 , ... , V Ek }, is increasing and l i m k ~ w k = w . By the preceding step,

(19) V W k = VVX{x:w~(~)>o } and IlWkllp < _ e R l l V V H p .

We have the a.e. pointwise convergence wk--+w and Vwk--+VvX{x:w(~)>o}. Further, w p is a m a j o r a n t to I W k - W l p and 1~Tvl p is a m a j o r a n t to IVWk--VvX{x:~(x)>O}] p.

Hence by the Lebesgue dominated convergence theorem, Wk--+W in Wol'P(O) and (17) holds. Passing to the limit in (19) we obtain (18). []

Definition. Let v E Wa,P(12). We introduce

- - R

M e v -~ A { v - v E : E is an a-cap set for v with diam E < R}

- ~ v - - V { v E : E is an a-cap set for v with d i a m E < R}, M n v = A { v - - v E : E is a c a p set for v with d i a m E < R}

~- v - V { v E : E is a cap set for v with diana E < R}

:a e R } .

T h e function 2~Rv is called the upper R-correction of v. Similarly we define the lower R-correction of v as

M Rv = V { v - - v E : E is a cup set for v with d i a m E < R}.

By L e m m a 5.3, v - - I ~ R v E W I ' P ( 1 2 ) ,

(20) V M R v = Vv)c(~:~/%(~)_,(x)},

- - R

and similarly for M a v and M Rv. We easily observe t h a t

- - R

(21) E is an a - c a p set for v with d i a m E _< R ~ M a v -- a a.e. on E.

It is less obvious t h a n it p e r h a p s seems to be t h a t the cap sets for the u p p e r R-correction of v with d i a m e t e r less t h a n or equal to R are cap sets for v and thus removed. Before we prove t h a t the u p p e r R-correction of v is R-capless ( L e m m a 5.6) we need some partial steps.

(23)

L e m m a 5.4. Let vEW~,P(~) and a E R . Then each a-cap set E for M R v with diam E<_R has measure zero.

Proof. Let E be an a-cap set for /~Rv with diamE<_R. T h e n M ~ v > a i.e.

on E. Obviously t ~ v = v i.e. on { x : . ~ v ( x ) > a } . Hence (v--a)XE=(t~RaV--a)XEE W~'P(~) and thus E is an a-cap set for v. Now (21) yields M ~ v = a i.e. in E, and hence [El =0. []

(22)

L e m m a 5 . 5 . Let v E W I ' P ( ~ ) and a E R . Then

> a i.e. on { x : > a}.

Proof. We write (23)

Consider s E R . We claim that

(24) w~ > a

If s>a, then obviously

- - R

wb = M b v , b E R , and w = MRv.

i.e. on {x: w~(x) > a}.

Thus by (21)

(25) w a = a a.e. o n { x E E : v ( x ) > a } . Passing to the lattice infimum, L e m m a 2.6 and (25) imply t h a t

wa-=a i.e. in {x:v(x) >max{a, Ws(X))}.

Since (x:wa(x)>a} D {x:v(x)>a} we conclude t h a t

w =v>wo>a i.e. on { x : w o ( x ) > a }

which proves the claim. Passing to the lattice infimum in (24) we obtain (22).

i.e. on E.

0<__ i v - a ) + <_v-s

[]

w~ > min{v, s} > min{wa, s } > a i . e . on ( x : wa(x) > a}.

T h e r e is nothing to prove for s=a. Suppose t h a t s<a. Let E be an s-cap set for v with d i a m E < R . Observe that { x E E : v ( x ) > a } is an a-cap set for v. Indeed, this follows from Corollary 2.4 as

(24)

268 Piotr Haj|asz and Jan Mal~

L e m m a 5.6.

Let

v E W I , P ( f i ) .

Then MRv is R-capless.

Proof.

We continue using the notation (23). Consider t E R and a t-cap set F for w with diam

F < R .

We claim t h a t if

a>t,

t h e n

(xEF:w~(x)>a}

is an a-cap set for wa. Indeed, L e m m a 5.5 yields the inequality

0 < m i n { ( w a - a ) X { ~ : w a ( ~ ) > a } , a - t }

< w - t

a.e. on F , and t h e n Corollary 2.4 implies t h a t

m i n { ( w a - a)X(x~F:w~(~)>~ } ,

a - t }

E W(~'P(fl).

Hence L e m m a 2.5 gives ( w a - a ) X { x E F : w ~ ( x ) > a } E W ~ 1,p

(fl),

which proves the claim.

Thus by L e m m a 5.4,

I{xeF:wa(x)>a}l=O. We

infer t h a t

w<_wa<__a

a.e. on F.

Since

a>t

was arbitrary, we conclude t h a t

w<<t

a.e. on F and thus

IFI=O.

T h e proof is complete. []

T h e next i m p o r t a n t step is the following lemma.

L e m m a 5.7.

If vEWI,P(~) is R-cupless, then MRv is R-cupless as well.

Proof.

We write

W --- M R V . Let F be a b-cup set for w with d i a m F < R . T h e n

O<_(b-v)+<b-w

a.e. o n F .

By Corollary 2.4,

(b--v)X{~eF:~(,)<b}

E W~'P(Y~) and

hence {xeF:v(x)<b}

is a b-cup set for v. Since v is R-cupless, it follows t h a t

(26) v > b a.e. on F.

Now, let E be an a-cap set for v with d i a m E < R . We claim t h a t

(27)

v - v E >b a.e. on FV~E.

To prove this, we distinguish two cases. If

a>b,

t h e n (27) holds as

v - v ~ ' = a > b

a.e. o n F O E .

Let us suppose t h a t a < b . T h e n by (26) and the fact t h a t

w<_v-vE-=a

a.e. on E we conclude t h a t

0 < b - a < min{(v-a)XE,

( b - - W ) X F } ~- min{v ~, --W F } a.e. on

FNE.

(25)

Since

min{v E, -- wF } XFnE = min{v E, --w E } E W~'P ( D ),

we get from Corollary 2.4 t h a t (b--a)XFAEEW~'P(~) which implies that [ F N E [ = 0 and (27) holds as well. By (26),

v - - v E : v > _ b a.e. on F \ E . This together with (27) yields

(28) v - v E >b a.e. on F.

Passing to the lattice infimum with the aid of L e m m a 2.6 we obtain w > b a.e. on F.

This shows t h a t I F [ : 0 . We have proved that w is R-cupless. []

Now we complete the proof of the whole theorem as follows.

Proof of Theorem 1.3. Let Rk"xa0. We set

W k = M R k u .

Write

E k : { X : W k ( X ) % U ( X ) } and E = f l E k .

k : l

Then from the definition of the corrections it easily follows t h a t for i > j we have u___< M R~ u___< M Rs u a.e., and hence neglecting sets of measure zero we get inclusions

By (20),

Hence

(29)

Now L e m m a 5.3 yields (30)

E1D E2D ....

Vwk = VuY~\Ek.

VWk--+ VuY~\ E in LP(ft) n.

tlw -ullp < CRk IlVull .

(26)

270 Piotr Hajtasz and J a n Mal~'

We deduce from (29) and (30) t h a t wk--+u in W I ' P ( ~ ) and Vu-= V u X n \ E a.e. in 12.

Now, we set

T h e n as above we obtain t h a t (31)

and

:--~Rj wk,j = M Wk.

V w k , j = VuX{x:~(x)=wk(x)=~k.~(~)}

(32)

lim IIWkj--Wklkp = O.

j - - ~

Hence we m a y find j ( k ) > k such t h a t

llwk,j(k) -wk]l = O.

Set

U k ~ W k , j ( k ) .

T h e n Uk converges to u in WI'p(f~), which is (c). From (31) we obtain (b). By the mirror version of L e m m a 5.6, Wk is Rk-cupless and by L e m m a 5.6 and L e m m a 5.7, uk is b o t h Rj(k)-capless and cupless. Hence, by L e m m a 5.1, uk is locally weakly monotone, and this is (a). T h e proof is finished. []

R e f e r e n c e s

1. ACERBI, n. and FUSCO, N., An approximation lemma for W l'n functions, in Material Instabilities in Continuum Mechanics (Edinburgh, 1985-1986) (Ball, J. M., ed.), pp. 1-5, Oxford Univ. Press, Oxford, 1988.

2. BALL, J., Convexity conditions and existence theorems in nonlinear elasticity, Arch.

Rational Mech. Anal. 63 (1977), 337-403.

3. BALL, J., Global invertibility of Sobolev functions and interpretation of the matter, Proc. Roy. Soc. Edinburgh Sect. A 8 8 (1988), 315-328.

4. BOJARSKI, B., Pointwise differentiability of weak solutions of elliptic divergence type equations, Bull. Polish Acad. Sci. Math. 33 (1985), 1-6.

5. BOJARSKI, B. and HAJLASZ, P., Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77-92.

6. BOJARSKI, B. and IWANIEC, W., Analytical foundations of the theory of quasicon- formal mappings in R n, Ann. Acad. Sci. Fenn. Set. A I Math. 8 (1983), 257-324.